Single Machine Vector Scheduling with General Penalties

Abstract

In this paper, we study the single machine vector scheduling problem (SMVS) with general penalties, in which each job is characterized by a d-dimensional vector and can be accepted and processed on the machine or rejected. The objective is to minimize the sum of the maximum load over all dimensions of the total vector of all accepted jobs and the rejection penalty of the rejected jobs, which is determined by a set function. We perform the following work in this paper. First, we prove that the lower bound for SMVS with general penalties is α(n), where α(n) is any positive polynomial function of n. Then, we consider a special case in which both the diminishing-return ratio of the set function and the minimum load over all dimensions of any job are larger than zero, and we design an approximation algorithm based on the projected subgradient method. Second, we consider another special case in which the penalty set function is submodular. We propose a noncombinatorial ee−1-approximation algorithm and a combinatorial min{r,d}-approximation algorithm, where r is the maximum ratio of the maximum load to the minimum load on the d-dimensional vector.

A Combinatorial 2-Approximation Algorithm for the Parallel-Machine Scheduling with Release Times and Submodular Penalties

“…Based on the primal-dual method, Liu and Li presented a 2-approximation algorithm for [16] single machine scheduling with release dates and submodular rejection penalty. More related results can be found in the surveys [17][18][19][20][21][22][23].…”

“…The vector scheduling problem [19,22,23] is a generalization of parallel machine scheduling, where each job J j is associated with a d-dimensional vector. Thus, the vector parallel-machine scheduling with release times and rejection penalties, which can be viewed as one generalization of the P|r j , reject|C max + π(R), deserves to be explored.…”

A Combinatorial Approximation Algorithm for the Vector Scheduling with Submodular Penalties on Parallel Machines

“…We propose a (2 − 1/m)(min r, d { })approximation algorithm, where r represents the maximum ratio between the largest and smallest components of the d-dimensional vectors among all jobs. Tis algorithm extends the fndings of prior results [17,20].…”

“…Clearly, if d � 1, the VSSP-PM problem is exactly the problem of parallel machine scheduling with submodular penalties studied in Liu and Li [17]. If m � 1, the VSSP-PM problem becomes the problem of single machine vector scheduling with general penalties studied in Liu et al [20].…”

“…Tey introduced a combinatorial 3-approximation algorithm to solve this problem. Liu et al [20] considered the single-machine vector scheduling problem with submodular penalties. Tey presented a combinatorial min r, d { }-approximation algorithm to solve this problem, where r represents the maximum ratio of the maximum load to the minimum load on the d-dimensional vectors among all jobs.…”